Colloquium: Friday, April 27, 2012

Leonid Bunimovich (Georgia Institute of Technology)

Title: FINITE TIME DYNAMICS


2:30 pm / CH 445

Abstract: In dealing with nonlinear dynamics people are used to its complexity
and ask questions only on their asymptotic in time properties. Answers to these questions occur to be still very difficult to obtain. In fact, even most of notions used in nonlinear dynamics (like Lyapunov exponents, decay of correlations, etc) already involve infinite time limit. I'll demonstrate that one could be more ambitious and some sensible questions on a finite time dynamics could be fairly completely answered. In fact, we even demonstrate new results on finite time evolution of Markov chains, which always were considered as simple approximations to chaotic dynamical systems. A major motivation for this new area of research came from the attempts to dynamically (rather than statically) characterize nodes and edges of dynamical networks.

Colloquium: Friday, April 20, 2012


Dongsheng Wu (University of Alabama at Huntsville)

Title: Regularity of Local Times of Fractional Brownian Sheets


2:30 pm / CH 445

Abstract: In this talk, we study the regularity of local times of fractional Brownian sheets. We provide sufficient and necessary conditions for the existence, joint continuity and smoothness (in the Meyer-Watanabe sense) of the local times. We also establish sharp local and global Hölder conditions for the local times. The main tools applied in our derivation are sectorial local nondeterminism of fractional Brownian sheets, Fourier analysis and chaos expansion of the local times.

This talk is based on joint works with Antoine Ayache and Yimin Xiao.

Colloquium: Friday, April 13, 2012


Qing Han (University of Notre Dame)

Title: Isometric embedding of surfaces in the 3-dim Euclidean space


2:30 pm / CH 445

Abstract: It is an old problem in geometry whether a surface (2-dimensional Riemannian manifold) admits an isometric embedding in the 3-dimensional Euclidean space. The local version of this problem goes back to Schlaefli in 1873. The first global result is due to Hilbert who proved in 1903 that the hyperbolic plane does not admit an isometric immersion in 3-dim Euclidean space. The first global affirmative result is due to Nirenberg in 1953 that any metric on the 2-dim sphere with a positive Gauss curvature admits an isometric embedding in 3-dim Euclidean space. In this talk, we will discuss these results among other results on local isometric embedding, including some recent results, and present some open questions.

Colloquium: Friday, April 6, 2012


Olivier Saut (University of Bordeaux, France)

Title: Data assimilation in tumor growth modeling: towards patient calibrated models using imaging devices


2:30 pm / CH 445

Abstract: Numerous mathematical models exist to describe cancer growth. Their main purposes are to dissect the various mechanisms involved in the disease, to evaluate or predict the growth or the effects of a therapy. Yet, in most cases, the quantitative results obtained through these models are restraint to simple setups or in-vitro studies. Indeed, these models have so many parameters (which are e.g tuning the interplays between the various phenomena influencing the disease) that it is practically difficult or even

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Colloquium: Friday, March 30, 2012


Weiyong He (University of Oregon)

Title: Frankel conjecture and Sasaki geometry


2:30 pm / CH 445

Abstract

Colloquium: Friday, March 16, 2012


Mate Wierdl (The University of Memphis)

Title: Subsequence ergodic theorems


2:30 pm / CH 445

Abstract: In this talk, I give an introduction to subsequence ergodic theorems.
After giving some motivation for the questions, I'll give an account
of recent results. We'll see some applications to other branches of
mathematics, such as combinatorics and number theory.  Throughout the
talk, several unsolved problems will be mentioned.  I keep the
discussion elementary, so it should be accessible to graduate
students.

Colloquium: Friday, March 9, 2012


Sergey Belyi (Troy University)

Title: On system realizations  of Herglotz-Nevanlinna functions


2:30 pm / CH 445

Abstract.

Colloquium: Friday, March 2, 2012


Klaus Schmitt (University of Utah)

Title: Nonlinear Elliptic PDE -- Some Dimension Dependent Phenomena


2:30 pm / CH 445

Abstract.

Colloquium: Friday, February 17, 2012


Éva Czabarka (University of South Carolina)

Title: Structural and enumeration results on some tree families relevant for bioinformatics


2:30 pm / CH 445

Abstract. Biologists use trees as a first approximation to represent the evolution of genetic material. These trees are ideally but not necessarily rooted, internal vertices have degree at least 3 (root may have degree 2) and the leaves are labeled with the names of the corresponding species. In case of phylogenetic trees the leaves are unique, in case of gene trees leaf-labels may repeat. I will present some results on phylogenetic and gene trees. These are joint work with various subsets of the collaborators P.L. Erdős, V. Johnson, A. Kupczok, V.Moulton and L.A. Székely.

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Colloquium: Thursday, February 16, 2012


Carmeliza Navasca (Clarkson University)

Title: Numerical Multilinear Algebra and Applications


9:00 am / CH 458
 
Abstract. Tensors are multidimensional array and thus, they are higher-order generalizations of matrices. In this talk, I will introduce tensors and tensor decompositions.  The idea that a tensor is decomposable into a sum of rank-one tensors was introduced by Hitchcock in 1927.  Multidimensional analogues of singular value decomposition surfaced in 1970's due to Tucker, Harshman, Carol and Chang.  Now new tensor decompositions and tensor based methods are making an impact in many application areas including problems in scientific computing that traditionally suffer from the "curse of dimensionality."  The application areas are in signal and image processing, chemical data analysis, compressed sensing and numerical methods for PDEs.

Colloquium: Friday, February 10, 2012


András Bezdek (Auburn University)

Title: How hard is to be fair when it comes to partitioning convex shapes?


2:30 pm / CH 445
 
Abstract. A convex partition of a polygon P is a finite set of convex polygons such that the interiors of the polygons do not intersect and the union of the polygons is equal to the original polygon P. The desire to create optimal partitions of a given convex polygon furnished a number of problems in discrete geometry. The properties used in optimization among others include equal area, equal perimeter and the number of pieces. The concept of fair partitions commonly refers to problems where simultaneously several quantities need to be kept equal. Variations of the cake-cutting problem are the most known problems among these. This talk surveys some of the 2D and 3D results and introduces some new variants. We are particularly interested in optimization problems which are restricted to triangulations only.

Colloquium: Friday, February 3, 2012


Cali M. Fidopiastis (UAB Department of Physical Therapy)

Geometry of Brain Space: New Paths for Rehabilitation?


2:30 pm / CH 445
 
Abstract. In this talk, we will explore the changing theories of how the brain processes information from a mathematical perspective. The main thesis of this discussion is that notions of how the brain processes information may be radically different from how we actually describe the outcome of that processing. For example, space perception may be seen as a problem for Euclidean geometry; however, current research supports that this process may actually belong to a class of Riemann geometries. Because Mathematics provides an objective means of operationalizing brain-processing phenomenon, elucidating the correct mathematical expression and constraints may allow for new directions in rehabilitation.

Colloquium: Thursday, February 2, 2012


Shannon L. Starr (University of Rochester)

Verifying physics predictions for frustrated constrained optimization problems


9:00 am / CH 458
 
Abstract. An optimization problem expressed as the sum of many terms is frustrated if the global optimizer fails to optimize some of the individual terms. One example we will consider in the talk is this: there is a committee with n members all of whom know each other. For each pair of members, they are either friends or enemies. A binary vote arises, yes or no, and the committee members want to vote along with their friends and against their enemies. But unless the friend/enemy graph is bipartite it will be impossible to find 1 vote which makes all pairs satisfied. The physicists discovered a method for dealing with such problems while studying spin glasses. We will describe our own contributions to the rigorous analysis of such problems verifying physics predictions.

Colloquium: Tuesday, January 31, 2012


Arnab Ganguly (ETH Zürich)

Stochastic simulation of biochemical reaction networks

10:00 am / CH 458
 
Abstract. Continuous time Markov chains are often used for
‘stochastically modeling’ chemical reaction systems. Although an exact
simulation of the state of the system is possible, it is often
computationally expensive for systems with high copy number of
reactant species. In this talk, I will discuss some approximation
algorithms, which make the simulation faster. The main focus of the
talk will be on a rigorous error analysis for the approximation
algorithms.

Colloquium: Thursday, January 26, 2012


Kun Zhao (University of Iowa)

Non Pattern Formation in Chemorepulsion

9:00 am / CH 458
 
Abstract. In contrast to diffusion (random diffusion without orientation), chemotaxis is the biased movement of cells/particles toward the region that contains higher concentration of beneficial or lower concentration of unfavorable chemicals. The former often refers to the attractive chemotaxis and latter to the repulsive chemotaxis. Chemotaxis has been advocated as a leading mechanism to account for the morphogenesis and self-organization of a variety of biological coherent structures such as aggregates, fruiting bodies, clusters, spirals, spots, rings, labyrinthine patterns and stripes, which have been observed in experiments. In this talk, I will present some recent development on the rigorous analysis of a partial differential equation model arising from repulsive chemotaxis which is a system of conservation laws consisting of nonlinear and coupled parabolic and hyperbolic type PDEs. In particular, global wellposedness, large-time asymptotic behavior of classical solutions to such model are obtained which indicate that chemorepulsion problem of this type exhibits strong tendency against pattern formation. The results are consistent with general results for classical repulsive chemotaxis models.

Colloquium: Friday, December 9, 2011


Vladlen Timorin (Moscow College of Economics)

Matings, captures and regluings

2:00 pm / CH 445
 
Abstract. Mating is an operation that produces topological models for rational functions out of polynomials (roughly by pasting together their filled Julia sets). In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components. (A joint project with I. Mashanova.)

Colloquium: Friday, December 2, 2011



Ergodicity of the Weil Petersson geodesic flow

2:00 pm / CH 445
 
Abstract. The Weil-Petersson metric is a Riemannian metric on the moduli space of a surface. It has negative curvature, but is incomplete. Analogy with the results of Hopf and Anosov for complete metrics of negative curvature suggested that the geodesic flow for the Weil-Petersson metric should be ergodic, but the incompleteness of the metric and insufficient knowledge of its geometry delayed a proof. We now know a great deal about the geometry of the Weil-Petersson metric, in large part due to the work of Scott Wolpert, and ergodicity of the geodesic flow has been proved by Burns, Masur and Wilkinson. The proof uses the results of Wolpert and the theory of nonuniformly hyperbolic dynamical systems, in the particular the work of Katok and Strelcyn.

Colloquium: Friday, November 18, 2011



Embedding graphs into the plane to approximate simplicial maps

2:00 pm / CH 445
 
Abstract.

Colloquium: Friday, November 11, 2011



From Sperner problems to mixed orthogonal arrays

2:00 pm / CH 445
 
Abstract. Sperner's theorem from 1928 asserts that one can obtain the largest number of subsets of an $n$-element set, such that none of them is contained by another, if one takes all subsets of size $\lfloor n/2\rfloor$ or   $\lceil n/2\rceil$, and there
is no other way to obtain the same number. The most elegant proof to Sperner's theorem is the Bollob\'as-Lubbell-Yamamoto-Meshalkin (BLYM) inequality, of which it is a simple corollary. Cases of equality have been characterized for the BLYM inequality.

Mixed orthogonal arrays are used by statisticians to design experiments in a balanced way. Construction of mixed orthogonal arrays of good properties is nontrivial and connected to the theory of combinatorial designs.

In the talk I'll review how generalizations of Sperner's theorem developed. The ultimate goal is to show a generalization of the BLYM inequality into inequalities, in which equality in all cases hold if and only if we deal with a mixed orthogonal array.

In addition, we construct a number of mixed orthogonal arrays using only the fractional part function.

This is joint work (in several papers) with (various subsets of) Harout Aydinian  (Bielefeld), Éva Czabarka (South Carolina), Konrad Engel (Rostock), and Péter L. Erdős (Budapest).

Colloquium: Friday, November 4, 2011



Boundary Data Maps, Perturbation Determinants, and a
Multi-dimensional Variant of the Jost and Pais Formula

2:00 pm / CH 445
 
Abstract. In this talk I will discuss boundary data maps (analogs of
the Dirichlet-to-Neumann map) and their connection to the Krein
resolvent formula, trace formula, perturbation determinant, spectral
shift function, and a multi-dimensional variant of the Jost and Pais
formula.

Colloquium: Friday, October 28, 2011



Limit theorem for dynamical systems

2:00 pm / CH 445
 
Abstract. One of the most important discoveries of 20th century mathematics is that deterministic systems can exhibit stochastic behavior. The stochasticity manifests itself in the fact that many classical limit theorems of probability theory are valid for statistics of orbits of dynamical systems with hyperbolic behavior. We present a selection of results in this direction and then discribe some of the recent advances dealing with non hyperbolic dynamical systems.

Colloquium: Friday, October 21, 2011



Some Remarks on the Second Wave During Flu Pandemics

2:00 pm / CH 445
 
Abstract. A striking characteristic of the last four influenza pandemics in the US had been multiple waves of infection. However,  the mechanism(s) causing  the second waves is uncertain.  Using mathematical models we exhibit two distinct mechanisms, each of  which can  account for the two waves in the US during the 2009 H1N1 pandemic.
Although the US had two waves of infections, China, which instituted strong border controls at the beginning of the outbreak,  had only one wave. Our models indicate that sufficiently strong border control in the US would have resulted in a single wave of infections, but with little difference in the total number of infected individuals.

Colloquium: Friday, October 7, 2011


Tangent cones and regularity of real hypersurfaces


2:00 pm / CH 445
 
Abstract. Can a real algebraic curve in the plane, i.e., the zero set of a polynomial of two variables, have "corners"? For instance, can a square be the zero set of a polynomial? Why not? In this talk we answer these questions and discuss a number of related results and applications. In particular we will see that there is a genuine geometric difference between the categories of real algebraic and  real analytic convex unbounded hypersurfaces of Euclidean space.

Colloquium: Friday, September 30, 2011


The set of packing and covering densities of convex disks


2:00 pm / CH 445
 
Abstract. (click)

Colloquium: Friday, September 23, 2011


Time-changed processes and Cauchy problems

Erkan Nane (Auburn University) 


2:00 pm / CH 445
 
Abstract. 

The link between concepts from probability and partial di erential
equations (PDE's) helped solve problems in analysis or nd easier and
shorter proofs for well-known results. Researchers have been fascinated
by these kinds of links. The classical well-known connection of a PDE
and a stochastic process is the Brownian motion and heat equationconnection.

In this talk, I will consider the Cauchy problems that can be solved
by running time-changed Markov processes. These are obtained by
taking Markov processes and replacing the time parameter with other
processes such as Brownian motion, symmetric -stable process of in-
dex 2 (0; 2], an inverse of a stable subordinator. We obtain frational
Cauchy problems or Cauchy problems involving the powers of the gen-
erator of the Markov Process by running these time-changed Markov
processes. In some special cases we obtain the equivalence of these two
types of Cauchy problems.

Colloquium: Friday, September 16, 2011


"Electric current in the presence of a Gaussian Thermostat"

Federico Bonetto (Georgia Institute of Technology) 


2:00 pm / CH 445
 
Abstract. I will review numerical and analytic results on a system consisting of one or many particles moving in a chaotic billiard under the influence on an electric field and a Gaussian thermostat.

Colloquium: Friday, September 9, 2011


A review about Lieb-Thirring inequalities

Michael Loss (Georgia Institute of Technology) 


2:00 pm / CH 445
 
Abstract. Lieb-Thirring inequalities are estimates of certain means of eigenvalues for Schr\"odinger operators in terms of the potential. They were invented by Lieb and Thirring in 1975 to give a simple proof of Stability of Matter. Because of its close connections to the calculus of variations and semi-classical analysis, calculating good constants in these inequalities has become a lively research endeavor with beautiful contributions notably by Lieb and Laptev -- Weidl. In this talk some of the ideas will be presented that led to recent progress in this field.

Colloquium: Friday, September 2, 2011


On the stability of a model of molecular networks

Hassan Fathallah-Shaykh (UAB) 


2:00 pm / CH 445
 
Abstract. The dynamics of molecular networks may lead to limit cycles, multistability, globally stable equilibria, or semistability.  Recently, the question of how the configuration/connectivity of a network influences its dynamics has attracted significant interest.  I will introduce a new model for molecular networks, related to the Lotka-Volterra and neural networks equations, and discuss new theoretical results on its global stability.  I will illustrate using network examples and numerical simulations and discuss biological relevance.

Colloquium: Friday, August 26, 2011


Flows with no minimal set

Krystyna Kuperberg, Auburn University


2:00 pm / CH 445
 
Abstract.  A dynamical system or a flow on a space X is an R-action, where R is the 
additive group of the reals. A nonempty, closed, invariant subset of X is minimal if 
it contains no proper, nonempty, closed, invariant subset. The set of problems
edited by P. Schweitzer contains the following question: Does there exist a flow with 
no minimal set? 2-dimensional surfaces of finite genus admit no such flows, but a 
surface with infinitely many handles admits a flow with no minimal set, see T. Inaba.
In a private conversation, P. Schweitzer asked whether a similar example exists 
on R3. The aim of this talk is to present a general construction of flows with no 
minimal set on

(1) R3,
(2) a 1-dimensional, non-compact, matchbox manifold,
(3) a 2-dimensional surface with infinitely many handles and infinitely many holes.